From the general perspec0ve

Transcription

1 Tensor field visualiza0on My view on the field January 2016 Geilo winter School Scien0fic Visualiza0on Linköping University From the general perspec0ve 2

2 to digging in the sand 3 Tensor field visualiza0on My view on the field January 2016 Geilo winter School Scien0fic Visualiza0on Linköping University

3 Tensors Why Do We Care? Tensors are everywhere (Simula0ons, physical theories) Can hardly be seen anywhere nobody cares? 5 Tensors Why Do We Care? If you re siwng at a cocktail party with a bunch of engineers, physicists, and mathema0cians, and you want to start a heated debate, Just ask out loud: What is a tensor? One person will say that, for all prac0cal purposes, a tensor is just a fancy word for a matrix. Then someone else will pipe up indignantly and insist that a tensor is a linear transforma;on from vectors to vectors. Yet another person will say that a tensor is an ordered set of numbers that transform in a par;cular way upon a change of basis. Other folks (like us) will start babbling about dyads and dyadics. [R. Brannon, Functional and Structured Tensor Analysis for Engineers, UNM Book Draft, 2003] 6

4 Tensors Why Do We Care? Vector visualiza0on Intui0ve no0on of a vector Tensor visualiza0on Intui0ve no0on? Prevalent Applica0ons Some driving ques0ons Long visualiza;on history, well accepted Many different applica0ons Only a few shared ques0ons No visualiza;on history, hardly known 7 Tensors Why Do We Care? Data?? Head Computer No common language across applica0ons But terminology mostly related to specific tensor invariants à Basis of our visualiza0on concept Eigenvalues Eigenvectors Questions Fuzzy Crisp Oeen scien0sts have no glue how to expect from the tensor data adapted from talk by Miriah Meyer given at Dagstuhl 2011?? 8

5 Tensors Why Do We Care? Tensor visualiza0on is field that is s0ll in its infancy (maybe except for Diffusion Tensor Imaging however this does not really happen in the visualiza0on community) 9 Overview I. Tensors As mathema0cal objects As physical descriptors II. Some basic visualiza0on methods III. Tensor Invariants for feature defini0on IV. The story of a collabora0on

6 I. Tensors as mathema0cal objects I Tensors as mathema0cal objects V n-dim vector space Second-order tensor T is a bilinear func0on Second-order tensor T as lin. operator T :V V! T (v,w) = w T T v, v,w V, T! n n T :V V T (v) = T v, v V v 1,v 2 V 2 T ( v 1,v 2 )! T = t 1n! "! # t nn t 11 t n1 This is not a tensor but a matrix Oeen a tensor is mixed up with its representa0on 12

7 I Tensors as mathema0cal objects The tensor is uniquely determined by its ac0on on all unit vectors Rota0on, deforma0on, reflec0on 13 I Tensors as mathema0cal objects Symmetric tensor Deforma0on, reflec0on no rota0on 14

8 I Tensors as mathema0cal objects S Posi;ve definite, symmetric tensor Deforma0on no rota0on, no reflec0on 15 I Tensors as mathema0cal objects Tensors can be analyzed using any convenient reference frame For specific reference frames, the tensor representa0on becomes especially simple à Basis given by the eigenvectors of the matrix Symmetric tensor Eigenvectors and eigenvalues Te i = λ i e i 16

9 I Tensors as mathema0cal objects There are different tensor Decomposi0on, which are oeen the basis for visualiza0on methods Polar decomposi0on stretch-rota0on Concatena0on of mappings 17 I Tensors as mathema0cal objects There are different tensor Decomposi0on, which are oeen the basis for visualiza0on methods Symmetric and an0-symmetric part Sum of mappings S = 1 ( 2 T + TT ), A = 1 ( 2 T TT ), s ij = 1 ( 2 t + t ij ji ). Symmetric part a ij = 1 ( 2 t t ij ji ), An0-symmetric part 18

10 I Tensors as mathema0cal objects There are different tensor Decomposi0on, which are oeen the basis for visualiza0on methods Isotropic scaling, anisotropic deforma0on (deviator) Many different measures for anisotropy available 19 I Tensors as mathema0cal objects Tensor Invariants En00es that do not depend on the representa0on Proper0es inherent to the tensor Examples Eigenvalues Determinant Trance All func0ons that only depend on the eigenvalues 20

11 I Tensor as physical descriptors I Tensor as physical descriptors Tensors are everywhere in Physics and Engineering Not because the world is linear But because linear is simple First order descriptors of the dependence of two vector fields à first term of the Taylor expansion v(w) = T w + higher order terms v,w They provide a more or less good approxima0on of the reality 22

12 I Tensor as physical descriptors Curvature tensor Gradient tensor Metric tensor Stress and strain tensor y σ y σz Diffusion tensor τ yx σ x z σ z σ y τ xy σ x x 23 I Tensor as physical descriptors Stress Tensor Geo- and Material Sciences 2 1. Solid block, with two applied forces 2. Implant design, stress simula0on in human bone 3. Notched block with external forces 1 3 Images: Kratz, ZIB 24

13 I Tensor as physical descriptors Metric, Curvature, Stress General Rela0vity Simula0on of gravita0onal field of a rota0ng black hole, respec0vely neutron star Images: Benger, Kratz, ZIB 25 I Tensor as physical descriptors Diffusion Tensor Medicine Imaging method: based on magne0c resonance tomography (MRT) measuring the diffusion of water molecules in 0ssues Applica0on: Reconstruc0on of neural fibers in human brain (tractography) Textured slice 3D Fiber tracking Glyph representations Images: Kratz, Breßler, Hotz, ZIB 26

14 I Tensor as physical descriptors Structure Tensor Image Analysis Image: Kratz, ZIB 27 I Tensor as physical descriptors Different Characters of Tensors - Examples Describes a deforma;on Posi0ve definite E.g. deforma0on tensor Determinant dett à volume change Describes a generator of a deforma;on Indefinite E.g. stress tensor (forces/ area) TraceT à volume change v' = T v v' = e t T v ( 1+ tt) v 28

15 I Tensor as physical descriptors Tensor Invariants Play a fundamental role in the understanding of applica0on specific tensors Every applica0on has its own invariants that are especially important. Thy oeen come with a domain specific language à Relevant invariants should guide the choice of the visualiza0on 29 II Some basic visualiza0on methods What has visualiza0on currently to offer

16 II Some basic visualiza0on methods Common prac;ce Color Representa0on of derived scalars, e.g. trace 2D slices and surfaces Stress simula0on, construc0on piece M. Stommel, Uni Saarbrücken B. Hirschberger (2004), Uni Kaiserslautern 31 II Some basic visualiza0on methods Local Methods - Glyphs Geometric objects represen0ng tensor characteris;cs Here: Ellipses Most frequently used visualiza0on The typical glyph: Ellipsoid 32

17 II Some basic visualiza0on methods Local Methods - Glyphs Many different glyph types in use When to use? Evalua0on of data quality Visualiza0on overview Probing Glyph design What should be represented? How should it be represented? Example from Material Sciences Glyph placement How many glyphs? Where to place them? [Schultz2010, Kratz2014] 33 II Some basic visualiza0on methods How should it be represented general design guidelines Local Methods - Glyphs Preserva0on of symmetry e.g. eigenvectors have no orienta0on, isotropic tensors have not dis0nguished direc0on Con0nuity similar tensors should have similar glyph representa0ons Disambiguate tensors with different values should be reflected by different glyphs [Schultz and Kindlmann 2010] Use glyphs that have been used in the community befor What should be represented applica0on specific guidelines Use applica0on specific invariants for the design 34

18 II Some basic visualiza0on methods Local Methods - Glyphs Resolving Percep0onal Issues Superquadrics [Kindlmann2004] 35 II Some basic visualiza0on methods Local Methods - Glyphs Resolving Percep0onal Issues Superquadrics [Schulz2010] 36

19 II Some basic visualiza0on methods Glyph placement How many glyphs? Where to place them? Local Methods - Glyphs Anisotropic sampling Images: Kratz, Bressler ZIB, Amira 37 II Some basic visualiza0on methods Local Methods - Glyphs Par0cle based and geometric methods on planes, in 3d, on surfaces [Feng2008] [Kindlmann2006] [Kratz2013] 38

20 II Some basic visualiza0on methods Tensor lines Integral lines similar to streamlines Follow one eigenvector field 39 II Some basic visualiza0on methods Tensor lines Direc0on field is not orientable. It is not a vector field!! 40

21 II Some basic visualiza0on methods Force applied to solid block Tensor lines Gravita0onal field of two colliding neutron stars Frey2001 Implant design Dick2009 Images: Zobel, ZIB 41 II Some basic visualiza0on methods Tensor lines Fiber tracking Tensors from diffusion MRI Tracking of neural fibers Major tensor lines can be used to approximate fibers Line tracing only for regions of high anisotropy 42

22 II Some basic visualiza0on methods Textures HyperLIC -2d kernel Fabric Texture Fabric Texture [Pang2003] [Hotz2004] [Eichelbaum2013] SplaWng glyphs [Benger2006] Sampling based textures Topology based textures Kratz2011 Auer II Some basic visualiza0on methods Textures Principal direc0ons and values Beam bend under its own load. Simula0on respec0ng evolving damage, 3 0me steps Data: Louise Kellog University of California. Data: Louise Kellogg, Department of Geology, University Ingrid of California, Hotz Images: Louis Feng UCD. 44

23 II Some basic visualiza0on methods Topological structure Segmenta0on of domain in areas of uniform direc0onal behavior Similar to vector field topology but different basic structures Topology 45 II Some basic visualiza0on methods Topology Topology based Segmenta0on Topology based Textures [Auer2013] Computa0on, Simplifica0on, Tracking [Tricoche ] 3D topology, cri0cal lines [Zheng2005] On Surfaces [Auer2012] Tensor field design [Zhang2007] 46

24 II Some basic visualiza0on methods Some technical stuff Glyphs from all perspec0ves Technical generaliza;on of vector field visualiza;on methods Textures What has been done so far? Much work on tensor processing Interpola0on / reconstruc0on Morphology Topology What is s;ll missing Applica0ons link form visualiza0on techniques to physical interpreta0ons No0on of features Ques0ons 47 III Tensor Invariants for feature defini0on Exploara;ve framework suppor0ng mul0ple applica0ons with different ques0ons à Domain specific feature spaces Structuring the data Manage complexity of the data Highlight trends Point at cri0cal/atypical behavior à Data atlas using a thumbnail like representa;on 48

25 III Tensor Invariants for feature defini0on Explora;ve framework suppor0ng mul0ple applica0ons with different ques0ons à Domain specific feature spaces Structuring the data Manage complexity of the data Highlight trends Point at cri0cal/atypical behavior à Data atlas using a thumbnail like representa;on 49 III Tensor Invariants for feature defini0on Tensor is uniquely defined by 3 eigenvalues à point in shape-space 3 eigenvectors à point in direc;on-space Tensor invariant Are the language of the applica0ons Use applica0on specific invariants to parameterize the shape space à shape descriptors I i (λ 1, λ 2, λ 3 ) à basis for the defini0on sta0s0c views, glyph design, similarity measure 50

26 The Stress Tensor and Failure Models Example for failure analysis in mechanical engineering à Mohr Circle τ 0 τ max c 51 III Tensor Invariants for feature defini0on Example Ordered shape space λ 1 λ 2 λ 3 52

27 III Tensor Invariants for feature defini0on Example Coulomb failure Parametrized by Mohr center c = λ 1 + λ 3 2 Max Shear τ = λ 1 λ 3 2 Shape factor R = λ 1 λ 2 λ 1 λ 3 [0,1] 53 III Tensor Invariants for feature defini0on Cluster in feature space Cluster representa0ves Reynolds glyphs Spa0al view: Top view of data 54

28 III Tensor Invariants for feature defini0on Cluster in feature space Cluster representa0ves Mohr circles Spa0al view: Top view of data 55 III Tensor Invariants for feature defini0on Clustering Mean shie Full feature space + spa0al coordinates Cluster sta0s0cs Cluster size Top-view of the two-point load Direc0onal histogram λ 1 -z y x 56

29 III Tensor Invariants for feature defini0on Clustering Mean shie Feature space (shear, shape) + spa0al coordinates Cluster sta;s;cs Cluster size Two-point load Spa;al-view Direc0onal histogram 57 III Tensor Invariants for feature defini0on Feature Space Selection Change of basis results in another set of shape descriptors Determines Exploration space Metric, similarity measure Glyph design 58

30 III Tensor Invariants for feature defini0on 59 III Tensor Invariants for feature defini0on 60

31 III Tensor Invariants for feature defini0on 61 Tensor Visualiza0on Driven Mechanical Component Design or The story of a collabora0on Kratz, Schoeneich, Zobel, Burgeth, Scheuermann, Hotz, Stommel, Pacific Vis 2014

32 IV Story of a Collabora0on Star0ng point Unspecific goals Different language 63 IV Story of a Collabora0on First Experiments Everybody shows what they have 64

33 Visualiza0on Framework Images: Kratz, ZIB, Amira 65 Engineering Product Development Process 66

34 Tensors in Material Sciences Engineering Product Development Process Adds a completely new perspective to the topic 67 Engineering Product Development Process Initial'structure' CAD7Model Structure4optimization Introduce4visualization4 using4tensor4lines to4support4this4process Constraints Requirement Speci1ications Design4rules Experience FE7'simulation validation Experimental' validation 68

35 IV Story of a Collabora0on First Experiments Everybody shows what they have 69 IV Story of a Collabora0on!"#$#%&'($)*+$*),' ,&!"#$%&'(#%$ )*+,(&*-*#%./*0(1(0'%("#$.%&,0%,&*5"/%(-(4'%("# 8'$*95"#5*:/*&(*#0*5 '#959*$(6#5&,3*$ 9-7'(#0*&%$#3" 1%&#2%$#3" 2($,'3(4'%("#5,$(#65%*#$"&53(#*$ %"5$,//"&%5%7* "/%(-(4'%("#5/&"0*$$ -./,)#0,"$%&' 1%&#2%$#3" 70

36 IV Story of a Collabora0on Hypothesis resul0ng from first experiments and discussions The tensor lines for a stress tensor field are related to the major load paths from the opera;ng loads to the fixa;on points of a technical part Tensor lines can be used to support the design of reinforcement structures 71 IV Story of a Collabora0on Design of a break lever Build reinforcement structure on basis of tensor line visualiza0on Combina;on of an automa;c analysis with the expert s knowledge Manageable but realis0c case Evalua;on Comparison to reference structure same volume Numerical valida0on Experimental valida0on 72

37 IV Story of a Collabora0on Design of the rib structure New geometries based on tensor visualiza0on No addi0onal steps in the workflow Stress Tensor Visualization Vis supported manual design CAD - Model 73 IV Story of a Collabora0on Numerical and experimental tests New designs (red, green, yellow) Reference geometry (blue) à All new geometries performed bexer as the reference geometry Rapid Prototyping and experimental validation: Test setup 74

38 IV Story of a Successful Collabora0on Advances in both fields Con0nuing Story Many exi0ng ques0ons and ideas Much fun Proposal submixed 75 Tensors as mathematical objects properties and theorems Tensor invariants as means for communication Visualization generic <-> specific Tensors as physical descriptors Relevance and meaning of mathematical properties and theorems is application dependent

39 Anisotropic Sampling of Planar and Two-Manifold Domains for Texture GeneraAon and Glyph DistribuAon, Kratz, Baum, Hotz, TVCG, 2013 Three-Dimensional Second-Order Tensor Fields: Exploratory VisualizaAon and Anisotropic Sampling (phdthesis), Andrea Kratz, 2013 VisualizaAon and Analysis of Second-Order Tensors: Moving Beyond the Symmetric PosiAve-Definite Case, Kratz, Auer, Stommel, Hotz, Computer Graphics Forum - State of the Art Reports, 2013 Tensor Invariants and Glyph Design, Kratz, Auer, Hotz, Visualiza0on and Processing of Tensors and Higher Order Descriptors for Mul0-Valued Data (Dagstuhl'11), Springer, 2014 hvp://